Vortex Rings of One Fluid in Another in Free Fall

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Vortex rings of one fluid in another in free fall

Nicholas Baumann, Daniel D. Joseph, Paul Mohr, and Yuriko Renardy

Citation: Physics of Fluids A 4, 567 (1992); doi: 10.1063/1.858328

View online: http://dx.doi.org/10.1063/1.858328

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Vortex rings of one fluid in another in free fall

Nicholas Baumann

3M Corp., Minneapolis, Minnesota 55455

Daniel D. Joseph and Paul Mohr

Department qfAerospace Engineering andMechanics, 107Akerman Hall, 110 Union St. SE, University of

Minnesota, Minneapolis, Minnesota 55455

Yuriko Renardy

Department of Mathematics and Interdisciplinary Center for Applied Mathematics,

Virginia Polytechnic

Institute and State University, Blacksburg, Virginia 24061

Experiments in which vortex r ings of one immiscible liquid are created in another from drops

falling from rest under gravi ty are presented and interpreted. These rings are associated with

circulations generated by viscosity and, unlike classical vortex rings which occur in miscible

liquids at high Reynolds numbers, they can exist even at very low Reynolds numbers. Since the

rings do not diffuse, they are well-defined. Nonetheless, there are many similarities in the

dynamics of formation and flow of miscible and immiscible rings. Parameters are identified

which appear to correlate the authors observations and photographs of some of the more

interesting events are shown.

I. CLASSICAL VORTEX RINGS

By way of comparison, it is instructive to recall that in

classical hydrodynamics, it is usual to consider vertical re-

gions embedded in an otherwise irrotational flow. In the case

of the ring, a cross section [Fig. 1 a) ] is like the potential

vortex [Fig. 1 (b) ] : the flow outside a cylinder which ro-

tates rigidly. This is the Taylor problem (flow between two

concentric cylinders with the inner one rotating and the out-

er one fixed) with the outer cylinder moved to infinity. The

streamfunction #of the flow, with (A, - +4x for the veloc-

ity in the x-y plane, and r = (x2 + y*) 2, is then $ = c In r.

Thus, A$ = 0 and the flow is irrotational. The potential vor-

tex satisfies the no-slip boundary condition at the cylindrical

boundary, and it is one of only a few potential flow solutions

of the Navier-Stokes equations.

Vortex rings can be generated in a number of ways. One

way is to impulsively eject a puff of fluid f rom a circular

opening into a bath of the same fluid, as in the smoke ring.

Another is to let a drop of liquid fall into a pool of the same

liquid.2 A third method is to force a buoyant fluid into a tank

of water (see Sec. 6.3.2 of Ref. 3) .4 These experiments do not

involve immiscible liquids. Rings are more easily created in

miscible rather than immiscible liquids. Thomson and

Newall did an interesting study of ring formation and their

stability in miscible and immiscible liquids. They stated the

following:

It is not every liquid, however, which, when dropped

into water, gives rise to rings, for if we drop into water

any liquid which does not mix with it, such as chloro-

form, the drop in consequence of the surface tension

remains spherical as it descends. In fact, we may say

that, wi th some few exceptions to be noticed later, rings

are formed only when a liquid is dropped into one with

which it can mix. This is important, because surface

tension has been supposed to play an important part in

the formation of these rings; it is difficult, however, to

see how any appreciable surface tension can exist be-

tween liquids that can mix, and as far as our experi-

ments go they tend to show that it is only the absence of

surface tension which is necessary for their produc-

tion.

On the whole, it is not surprising that a phenomenon

which occurs at zero interfacial tension also occurs at small

values of interfacial tension. And small here means with re-

spect to viscous effects, so that the actual numerical value of

the coefficient of interfacial tension does not have to be

small. The processes are similar whether miscible or immis-

cible liquids are involved, up to the ring stage. However,

interfacial tension does affect the breakup pattern: for exam-

ple,6 it can prevent the smaller-sized drops from repeating

the sequence of ring formation and breakup; this limits the

vortex cascade (see Figs. 1 and 2 of Ref. 5 for a description of

this cascade for the case of an ink drop in water) to one or

two stages. Membrane

rupture

is another form of breakup.

The rupture strength or breaking strength of membranes is

not well understood but it may be related to surface tension.

We know that the rupture strength generally decreases with

surface tension. When the surface tension parameter is small

but the surface tension is large, a vortex ring spanned by a

tough permanent membrane can form. In other cases in

which interfacial tension has been reduced by surfactants,

the membrane is blown out and an unstable vortex ring of the

type shown in our photographs forms. The difference be-

tween strong and weak interfacial tension is illustrated in

Figs. 2 and 3. In Fig. 2, a smaller drop of 1000 cS silicone oil

is sucked into the wake of an oblate ringlike cap of the same

silicone oil falling in a contaminated soybean oil. The mem-

brane in this system is too tough to break. On the other hand,

if a surfactant (97% dye, 3% Rhodamine B base powder,

Aldrich Chemical Co., Milwaukee, WI) is added (as in Fig.

3), the membrane breaks readily. Figure 1 d) of Ref. 7 illus-

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(c)

(b)

FIG. 1. (a) The two circular cross sections of a ring vortex are shown. The

flow is as ndicated. The dashed line denotes the axis at the center ofthe ring.

The ring as a whole turns about this axial line. The continual turning of the

ring is analogous to the rigid-body rotation of a straight cylinder shown in

(b) if the ring were cut and straightened. (b) An infinitely long solid cylin-

der is rotating with azimuthal velocity Rr, where fi is the angular speed.

The flow outside the cylinder is irrotational and given by the streamfunction

*=clnr.

trates this well. The existence of a spanning membrane in

miscible liquids is hard to understand without invoking the

idea of transient inter-facial tension induced by momentarily

strong gradients of composition. Such notions were intro-

duced by Kojima et aL8 and are considered in more detail in

Ref. 9.

II. THE NORMAL STRESS BALANCE

It is probable that the.parameters which control the de-

formations of drops to rings in free fall are associated with

the stress balance at the interface:

-~~n+2ED[u]jln+S*n2H=O,

where His the mean curvature at a point on the interface Z

and D [ u] is the rate of strain. This equation may be decom-

posed into normal and tangential parts.

We denote

I[*n = (.>d - (*),,

to be the jump in the quantity - across the interface, where

the subscript

d

stands for drop and o stands for the outer

fluid. In the equilibrium case (that is, no flow), the drop or

bubble pulls into a sphere with radius

R

r =

R2

= a and equi-

librium pressures satisfying

l[pq =p; -pz = W*/a.

By using the condition that the velocity is continuous at

the interface and the continuity equation, the reader may

verify that

mn = 09

(1)

where we denote

D,,

= n*D[u]*n.

The next step in the reduction of the normal component

of the stress balance on the interface B is the decomposition

of the pressure into an equilibrium partp, a hydrostatic part

ps = pgz, with p = pdg.z in the drop, and a dynamic part Il

due to the motion. We use a coordinate system where the

origin is the center of the spherical drop, and we denote the

parametrization for the surface of the drop by z = z, (x,y) .

Thus, bq = b]gz,, and

(b)

(cl

.

FIG. Z~~ailure of coke-through of captured drop of 1000 CSsilicone oil in

an indented oblate drop of the same silicone oil falling through contaminat-

ed safflower oil. (a) The captured drop i s sucked strongly into the wake

behind the oblate drop. There is a tail drawn out of the captured drop by the

motion of safflower oil in the wake which is reminiscent of the tail behind

drops in miscible liquids (cf. Fig. 11). (b) The drops are sucked into strong

contact. (c) The captured drop decelerates under the restraining action of

the silicone oil membrane on the oblate drop which never breaks.

Combining now ( 1) and (2) with the stress balance equa-

tion at the interface, we find that

--Pn+~,,M+IGOn@~+~* -+++-$)

(

1 2

= 0.

(3)

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Baumann et a/. 568

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(a)

(b)

(d)

(4

FIG. 3. Ring formation in 1000 CS silicone oil with a surfactant falling in

soybean oil. The surfactant is a trace amount of 97% dye with 3% Rhoda-

mine B base powder. (a) One indented oblate sphere accelerates in the wake

of another; (b) they come close. (c) Poke-through: the large ring loses its

membrane. (d) The small ring never pokes through; it retains the oblate

indented shape. (e) Beginning of the two-lobe instability of the Rayeligh-

Taylor type. (f) The instability can be compared with Fig. 6(b) where the

membrane does not break.

The dynamic pressure is of course an unknown which

must be determined from the solution.

III. DIMENSIONLESS PARAMETERS

To identify dimensionless parameters, we scale lengths

with a, the radius of the equivalent spherical drop or bubble

with the same volume, and velocity with U to be specified

later (see next section). The normal stress balance at the

interface (see previous section) shows that there are four

forces at work: gravity, surface tension, inertia, and viscos-

ity.

The viscosity ratio

M=K-l/p,

(4)

where,zu, is the viscosity of the drop and,u is the viscosity of

the ambient fluid, is very important.

The ratio of inertia to viscous forces is measured by the

Reynolds numbers

R = lJa/v, R, = Us/v,,

(5)

where Y =&p, Yd =pud/,od. To form immiscible vortex

rings, inertia is important because the drop will be close to a

sphere if

R

and

R,

are sufficiently small. lo

The viscous part of the normal stress in the drop is

scaled by

U&/a

and in the exterior fluid by

Up/a.

The inter-

facial tension term in the stress balance is scaled by 5 */a.

The ratio of the stress associated with inter-facial tension to

the viscous part of the normal stress in the outer fluid is

S*/,LLU= J/R,

(6)

where

J = S *a/tip

(7)

is Chandrasekhars capillary number (used in his study of

the capillary instability of a jet; see Sec. 111 of Ref. 11) for

the outer fluid. Similarly, for the inner fluid we have

S*/,ud U = Jd/Rd,

(8)

where

MJd/Rd = J/R.

(9)

For a ring to form, the tendency for interfacial tension to

keep the drop spherical should be overcome by the effect of

viscosity to distort it. Thus, we expect to-see rings when

J/R 4 1. In our experiments, we got ring formation only

when M# 1. We did not observe rings in immiscible liquids

when A4 < 5. The condition MS 1 may not be universal. Cer-

tainly, it is easier to form vortex rings in miscible liquids; for

these, J/R = 0, but evidently when Mz 1 it is possible to

form vortices from ink drops falling in water.

We have already mentioned that inertial effects are al-

ways important in deforming the drop away from a sphere.

These effects can be measured by the Weber number, the

ratio of interfacial tension to inertia. The inertial part of the

dynamic pressure for the outer fluid is scaled with p U 2 and

the drop withpd U2. The ratio of interfacial tension to inertia

in the outside fluid is

(s*/a) J 1

-=--=-,

pu2 R2 w

and in the drop is

(S*/a) Jd 1

-=- =-,

pciU2 R; W,

where

Wis

the Weber number. Obviously,

(10)

(11)

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TABLE I. Fluid properties. Table notes: Canola oil is also termed rapeseed oiI;the glycerin listed is 0.99 pure USP glycerin; the percentages listed for golden

syrup and glycerin are dilutions with water; Alconox is an industrial glass cleaner and is used as a surfactant with water.

Pluid

Canola oil

Glycerin

0.95 Gly

0.9 1 Gly

0.92 golden

syrup

Lyles golden syrup

Motor oil 30W

Olive oil

Palmolive

soap

Safflower oil

Sesame oil

Shell Research oil

Silb 5 CS

Sil 100 cS

sil200 cs

Sil300 CS

sil400 cs

sil500 cs

Sil600 CS

Sill000 cs

sil 10 000 cs

Sil30 Ooo cS

Soybean oil

Water

Water + Alconox

Walnut oil

Density

Kinematic viscosity

Viscosity

(g/cm3)

(CS)

(g/cm set)

0.915

67

0.61

1.265

656

8.29

1.245

244

3.03

1.240

113

1.40

1.400

2 606

36.49

1.440

20 804

299.58

0.886 316 2.80

0.914

69

0.63

1.05

238

2.50

0.920

51

0.47

0.920

64

0.59

0.895

2 037

18.23

0.930

5

0.05

0.960

100

0.96

0.970

200

1.94

0.970

300

291

0.970

400

3.88

0.971

500

4.86

0.971

600

5.83

0.971

1000

9.7 1

0.975

10 000

97.50

0.975

30 000

292.50

0.922

53 -~

0.49

1.000

1

0.01

1.080

33

0.36

0.925

51

0.47

a0.95 Gly = 95% glycerin in 5% water.

Sil= Silicone oil with indicated viscosity.

pd Jd J

--=-*

p Rd R2

(12)

Since p and pd do not differ greatly in our experiments, the

Weber number is nearly the same in the outside fluid and the

drop. The Weber number W, for systems that do form rings

ranges between 330 and 9600 whereas the W, for systems

that do not form rings ranges between 0.3 and 11000. The

low Weber number drops are spherical.

IV. PHYSICAL AND OTHER PROPERTIES

The physical properties are density, viscosity, and inter-

facial tension. Other properties used in our discussion are the

velocity U and the drop size a. First, we discuss the fluid

properties.

Table I lists the fluids used in the experiments. The den-

sities were measured using a Curtin Scientific hydrometer at

approximately 21 C. The viscosities were measured using

standard Cannon Fenske tube viscometers. The interfacial

tension S * was measured with the spinning rod tensiometer.

Our early experiments on vortex rings were carried out

in a Plexiglas box 3 in. square and 8 ft long. The top of the

box is open to allow introduction of the drop and the bottom

is closed by a valve. The valve holds the host fluid in and

allows the removal of the dropped fluids that collect at the

bottom. The Plexiglas is clear to allow good visualization

and photographic recording. The apparatus is backlighted

by retlecting incandescent light of f of a translucent Plexiglas

sheet. The most recent experiments were carried out in a

tube, which differed from the previous apparatus in that it is

made of glass and has a circular rather than square cross

section. The tube is 4 ft long, and like the Plexiglas box, it is

open at the top, and closed at the bottom with a valve.

The method for introducing the drop into the vortex

ring box is as follows. A 10 cm3 drop of the more dense fluid

was carefully placed on top of the host fluid with a calibrated

beaker. This gives

$-a3 = 10 cm3 or a = (2.39) 3 cm = 1.34 cm.

Care was taken to ensure that the drop was not splashed or

accelerated into the host fluid.

A parametric study of drop size was also carried out

with volumes other than 10 cm3. The results of these studies

are summarized in Fig. 4.

The velocity scale we use to calculate the Reynolds

numbers is U given by (see Sec. 4.9 of Ref. 12)

u=+$,,, -p)(s). (13)

Using this U, R, = ua/vd depends only on measurable

quantities and may be interpreted as the ratio between the

buoyancy and viscous forces. The assumptions leading to

( 13) are that the fluid is a sphere falling at constant speed in

Stokes flow, and that if the shear stress and velocity are

matched at the interface, then the normal stress is automati;

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FIG. 4.

-2 4 6 8 10 12 14

VOLUME (cc)

Distance traveled by a falling drop before a vortex ring forms as a function of drop volume. The formation of the ring occurs

spanning the ring breaks.

tally matched. The analysis does not say anything about

what the drop would do if it were not falling at the terminal

speed; for example, in the experiments, the drop starts at

rest, some drops do not attain any steady speed, and more-

over appear not to reach the speed predicted by this formula.

It is difficult to decide

apriori

on a velocity scale because we

do not have a formula for predicting the velocity as the drop

changes shape. For each experiment, one could measure the

maximum speed attained by the drop and use that as U, and

this t ype of data is available for some of the experiments.

The velocity of the drop as it falls in the vortex ring box

has been measured for some situations and found to be much

smaller than the value from ( 13). Measurements of the ve-

locity of a falling drop were made by recording the time it

took for the drop to cross a 6 in. region of the box. Five such

regions were selected to best capture the rate of fall at critical

sections. The records were taken ten times for each region

and the average velocity was calculated. Figure 5 shows the

average velocity versus distance down the tube for three

types of glycerin (e.g., 90% glycerin means 90% glycerin in

10% water) and silicone oil falling in soybean oil. (Rings

were observed in the 100% glycerin case, but not in the other

two cases, which happen to have higher velocities in the fig-

ure.) Take, for example, the data in this figure. Compared

with this, the value of U from ( 13) is approximately 110

in./sec, which is about 20 times the actual average velocity.

This is consistent with the notion that a spherical drop

would fall faster than a flattened spheroid or a ring. The

swings in the measured speed reflect the changes in the shape

of the drop as it evolves into a ring and decays. We should

the membrane

therefore keep in mind when looking at the tables that the

true Reynolds numbers are probably an order of magnitude

less than those tabulated.

There are also situations where U from ( 13) turns out to

be large, which is inconsistent with one of the assumptions in

the derivation of ( 13); but since Stokes drag is less than the

actual drag at higher Reynolds numbers, we expect that the

U is an upper bound on the actual maximum velocity. Thus

our tabulated values of Jd/Rd and J/R in the sequel underes-

p 100% lycerin

L.

f.

4-

1

:

L

3

e

m

1

>

m

2

i-

- 500cS. silicone

1

10

20

30

dbtance down lube (inchesi)

1

40

FIG. 5. The average velocity versus distance down the vortex ring tube for

the designated liquids falling in soybean oil.

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timate the importance of surface tension, but consistently, so

that they should probably be an order of magnitude larger

than they are. This would imply that the switch in the behav-

ior from ring formation to no ring is actually occurring at a

value of J/R of order 1. This, in fact, is what one would

expect.

It is interesting that the condition for ring formation (on

M and J/R) appears to hold for the entire wide range of

Reynolds numbers encountered in the experiments. Why? In

the normal stress condition at the interface, the only term we

have not really commented on above is the pressure term,

which is multiplied by the Reynolds number R. It appears

that this term does not affect the ability to give birth to a ring:

indeed, the factor R appearing there can be made to disap-

pear just by changing the way the pressure is nondimension-

alized.

The formation of vortex rings always involves the break-

ing of a membrane, by poke-through or blowout, and the

breaking strength (toughness) of a membrane is very diffi-

cult to control, especially in silicone-vegetable oil systems.

Our early experiments were recently repeated with good suc-

cess except for the breaking of silicon-vegetable oil mem-

(a)

(b)

FIG. 6. 1000 CS silicone oil falling in contaminated safflower oil. (a) A

vortex r ing with circulat ion has developed but a tough membrane spans the

ring. (b) The ring is unstable in the usual way (Rayleigh-Taylor in stabil-

ity) forming the characteristic drops [cf. Fig. 3 (d) 1, but the membrane

breaks.

572


FIG. 7. Vortex ring of 1000 cS silicone oil with trace amounts of surfactant

(Igepal) falling in soybean oil after blowout.

branes. Some experiments were carried out with a contami-

nated safflower oil with various additives. We could never

break a membrane in a silicone-contaminated safflower oil

system (Figs. 2 and 6). The breaking strength of a mem-

brane may be related to interfacial tension since we could get

tough membranes to break by adding certain types of surfac-

tants to the silicone oil (trace amounts of 97% dye, 3% Rho-

damine B base powder in Fig. 3; trace amounts of Igepal in

Fig. 7). We also had difficulty breaking membranes in a

silicone-soybean oil system, even when uncontaminated

fluids were used. However, the oils used in the most recent

experiments were not exactly the same as those used earlier,

and it is possible that the newer oils had an interfacial tension

large enough to prohibit vortex ring formation. As was the

ease for the contaminated-oil system, rings were formed

when the above-mentioned surfactants were added to the

silicone oil.

The low values of surface tension in the silicone oil-

vegetable oil systems may indicate the possibility of small-

scale activity at the interface. This activity could affect the

boundary conditions involved, but more research must be

done before any definitive statements may be made.

V. DlSTORTlON OF THE SPHERICAL DROP

It is well known that a spherical drop or bubble, moving

slowly, in Stokes flow can keep a spherical shape even when

interfacial tension is suppressed.1213There is numerical and

experimental evidence to suggest that these solutions are sta-

ble to small disturbances i4, but not to large disturbances.

When viscous effects win over the effect of inter-facial

tension, a falling drop cannot maintain a spherical shape.

Numerical solutions have been obtained16 for steady stream-

ing flow past an axisymmetric drop over a wide range of

Reynolds numbers, interfacial tension, viscosity ratios, and

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density ratios. Their results indicate that at lower Reynolds

numbers, the shape of the drop tends toward an indented

oblate shape with decreasing interfacial tension, and at high-

er Reynolds numbers the drop becomes more disk shaped

with decreasing interfacial tension.

In our experiments, the drop is released at zero speed

and undergoes accelerations and decelerations, so that the

results mentioned above concerning steady motions cannot

strictly be used to infer anything about what our drop is

doing. Moreover, as mentioned in the previous sect ion, mea-

surements of the drop speed indicate that it of ten does not

reach the steady speed predicted by the formula ( 13 ) . How-

ever, there are similarities with these analyses and what we

have seen.

Figure 8 shows an indented oblate drop like those com-

puted in Ref. 15 at low Reynolds numbers (see, for example,

their Fig. 3). Experimental observations suggest that the

streamline pattern on the concave side of the cap is probably

like that of Fig. 9; there are no points of separation or vorti-

ces in this guess about the underlying fluid dynamics. The

(b)

I.

FIG. 8. Indented oblate drops falling in safflower oil. (a) Water M = 0.02,

J- 4534 S * = 3.39 dyn/cm. Indentation never develops in water and oil

systems without surfactants. (b) 500 cS silicone oil. ihese are the most

common shapes when falling. The high viscosity drop develops a circulation

that brings it closer to a vortex ring.

(a)

i

FIG. 9. DeveIopment of vorticity in a drop falling from rest. The stream-

lines are sketched in a frame moving with the drops: (a) from experimental

observations at sufliciently small velocity; (b) larger velocity.

suction in the cap, call it a wake, is large and small drops and

even large drops are easily captured by the indented drop, as

shown in Figs. 2(a), 10, 11, and 12(a). Ifthe conditions are

right, the drop in the wake will poke through the membrane

spanning the indentation, as in Figs. 3 and 12, but if the

membrane is tough, as n Figs. 2 and 6, the poke-through will

fail.

VI. FORMATION OF RINGS

Stuke performed experiments like those reported here.

He cites the work of Northrup* who used paraffin in a water

bath, where interfacial tension is large. Northrup needed to

inject the paraffin at high speed so that viscous forces would

create a circulation of sufficient magnitude, as in the case of

air injected into water. The larger the interfacial tension, the

faster the intrusive speed necessary to create a ring. Stuke

showed that rings would form at slower speeds when water

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(a)

(b)

FIG. 10. Spheres nested in the wake ofan indented oblate drop. (a) Glycer-

infallinginsoybeanoil M= l&9,5 = 0.45, S* = 18.45 dyn/cm. (b) 500~s

silicone oil in contaminated safflower oil.

was replaced with amyl alcohol, with a consequent lowering

of interfacial tension. The slower speeds allowed these pro-

cesses o be recorded in photographs which can be compared

with the photographs of this paper. Initially, there is a mem-

brane across the hole of the ring (cf. Figs. 6 and 12) and then

the membrane ruptures. Once formed, the ring starts to ex-

pand rapidly and the bulk of the paraffin flows into two or

three bulges around the ring (cf. Figs. 3 and 7). These hea-

vier bulges fall faster, so that the ring bends and breaks into

two or three drops [see his Fig. 5, Fig. 2(b) of Ref. 6, and our

figures]. If a drop were large enough, it would form another

vortex ring and the sequence repeats itself. An analogous

description of the ring instability for miscible liquids at slow

speed is given in Fig. 4 of Ref. 8 and in Ref. 7.

If the conditions are right, if the drop is much more

viscous than the host fluid CM% 1) and the ratio J/R of

interfacial to viscous forces is not too large, then the spheri-

cal drop will evolve toward a vortex ring. The entries in Ta-

bles II-IV for drops of silicone oils in soybean oil exemplify

these effects well. The viscosity of silicone oils can be varied

through careful mixing without changing their density or

surface tension appreciably. It was observed that when the

viscosity of the drop was lower than a certain value (about

500 CS here), rings did not form. In particular, when the

FIG. 11. A streamline pattern for Fig. 2. The flow in the wake could pull out

a tail f rom the nested sphere if the wake were strong as in Fig. 2(b), or the

surface tension weak as in the case of miscibl e li quids.

drop was less viscous than the bath, even with very low inter-

facial tension, rings did not form, as in Fig. 13.

Inertia alone will not cause a ring to form. Indented

oblate drops like those shown in Fig. 8 are the most robust of

the falling drops. If conditions are such that the viscous ac-

tion of the host fluid can create a permanent circulation [like

that sketched in Figs. 9 (b) and 111 of sufficient strength, the

drop will begin to look like a ring, spanned by a membrane,

as in Figs. 10, 11, 12 (a), and especially 6 (a). A free ring will

form only if the membrane breaks. The membrane may or

may not break. If it breaks, it does so either by poke-through

of a smaller drop caught in the wake as in Figs. 6 and 12, or

by blowout. Blowout can best be understood by the failure of

blowout shown in Fig. 6 (a). Blowout can occur only if the

membrane is very weak as in miscible liquids or in low inter-

facial tension systems like those shown in Figs. 7 and 14.

Vortex rings are unstable; whether or not the membrane

has broken, the ring will expand rapidly. The rapid extension

is a universal characteristic of the instability. If a membrane

remains and no drop rests in the wake to poke through, the

membrane will stretch and either rupture or fold as in Fig.

6(b).

Because of capillarity, draining, or other causes, bulges

develop on the ring; these fall faster than the rest of the ring,

and fluid drains rapidly to the heavy bulges, exacerbating the

instability. This instability can be considered as a manifesta-

tion of the Rayleigh-Taylor instability of the heavy fluid into

the light, when the heavy fluid has the shape of a vortex ring.

In our experiments, the draining almost always occurs at just

two points of the ring, more or less at opposite points on the

ring as in Figs. 3 (f) and 12(d), and in the figures of Refs. 7

and 8. This type of instability scenario can occur even for

ringlike structures like the one shown in Fig. 6 (a) in which

the membrane does not break and it leads to the folded ring

shown in Fig. 6 (b) . The heavier places fall faster and the

ring bends and breaks into drops.

OBrien, I9 in reviewing her own work and that of Ref.

17, noted that the number of bulges which develop on the

ring depends on the Reynolds number and is two for Reyn-


Baumann eta/.

574


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(d)

FIG. 12. Poke-through of 1000 cS silicone oil in safflower oil M = 19.8, J= 0.03, S* = 2.41 dyn/cm. (a) Silicone oil spheres nested in the wake of an

indented oblate drop of the same oil. (b) Poke- through leads to a vortex ring. (c) Vor tex ring ( Rayleigh-Tay lor) instability is the rapid expansi on of the ring

diameter and the draining of the oil into the falling bulges. (d ) Two new indented oblate drops form f rom the falling bulges in a replication of the dynami c

sequence.

TABLE II. Systems that form vortex rings. Gly denotes 100% glycerin; 0.95 Gly denotes 95% glylcerin in 5% wafer; Shell denotes Shell Research oil; Sil

denotes silicone oil with the indicated viscosity ; Soy denotes soybean oil. The difference&denotes (density of dropped fluid) - (density of host fluid); Av

denotes (viscosity of dropped fluid) - (viscosity of host fluid): this difference is negative in some of the systems in the tables.

System

AP

(g/cm)

S*

u

(dydcm 1

(cm/set)

Rci

t&/P

M

J,/Rci

J/R

Gly/Soy

0.343 606

18.45 278.5 57

1.37 16.9

0.008 0.135

Gly/canola

0.350 589

35.78 227.5 46

1.38 13.6

0.02 0.272

500 cS SWcanola

0.056 433

2.33 37.0

10 1.06 8.0

0.01 0.080

Gly/olive oil 0.35 1 588 lt3.50 242.0 49 1.38 13.2 0.005 0.066

500 cS SiUolive oil

0.057 431

3.10 58.0

16 1.06 7.7

0.01 0.077

Gly/safflower

0.345 606

15.65 291.5 59

1.38 17.6

0.006 0.106

500 cS SiVsafflower

0.05 1 450

7.44 43.6

12 1.06 10.3

0.03 0.309

Gly/walnut oil

0.340 605

39.51 285.8 58

1.37 17.6

0.02 0.352

500 cS Sil/walnut oil

0.046 449

4.15 39.1

10 1.05 10.3

0.02 0.206

Gly/sesame oil

0.345 592

15.20 216.3 44

1.38 14.1

0.008 0.113

500 cS SiNesame oil

0.051 436

3.40 34.9

9 1.06 8.2

0.02 0.164

500 cS Sil/Soy

0.049 447

1.68 40.3

11 1.05 9.9

0.008 0.079

600 cS Sil/Soy

0.049 547

2.68 40.1

9 1.05 11.9

0.01 0.119

1000 cS SiVSoy

0.049 947

2.41 39.7

5 1.06 19.8

0.006 0.119

10 000 cS SiVSoy

0.953 9 947

3.29 42.3

0.6 1.06 199.0

0.0008 0.159

30 000 c3 SiVSoy

0.053 29 947

6.49 42.3

0.2 1.06 597.0

0.0005 0.298

Golden syrup/Soy

0.518 20 751

42.20 413.3

2.7 1.56 611.4

0.0003 0.183

0.92 Golden syrup/Soy

0.478 2 553

28.46 382.6

19.6 1.52 74.47

0.002 0.149

Palmolive/Soy

0.128 185

18.10 108.0 61.0

1.14 5.10

0.0006 0.003


Baumann et al.

575

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TABLE III. Systems that do not form vortex rings.

System

A/J

( g/cm3 )

Av

s*

u

(CS)

(dyn/cm)

(cm/=)

Ri

Pd~P

M

Jc,/&

J/R

0.95 Gly/Soy 0.323

0.91 Gly/Soy

0.318

WatedfOW motor oil

0.114

Water/Shell 0.105

Gly/Shell 0.370

500 cS Sil/Shell 0.076

5 CS Sil/Soy 0.608

100 cS Sil/Soy 0.038

200 cs m/soy 0.048

300 cS SiVSoy 0.048

400 cs sillsoy 0.048

Water/Soy

0.078

Water + Alconox/Soy

0.158

0.60 Golden syrup/Soy 0.342

194 13.43

269.8 148

1.35 6.2

0.02

0.124

63

11.49 277.3 328 1.34

29

0.03

0.87

315

9.22

23.7 3172 1.13

0.004

38.85

0.156

2036 42.14 3.4 450

1.12 o.OCQ5

1251

0.625

1381 27.83 10.3 2

1.41 0.45

0.34

0.153

1537 5.82

2.2 0.6

1.08 0.27

0.53

0.143

- 48

1.14 9.2

245 1.01

0.10

2.7

0.27

47 2.75 34.1 46 1.04 2.0 0.08 0.16

147 2.16 40.9

28

1.05 4.0

0.03

0.12

247

2.71

40.2 18 1.05

5.9

0.02 0.118

347 2.67 39.7 13 1.05

7.9

0.02

0.158

- 52

3.39

92.3 12 337 1.08

0.02

0.37

0.074

- 20

4.64

156.0 632

1.17 0.73

0.08

0.056

- 27.5

7.42 341.0

1788

1.37 0.659

0.07

0.046

olds numbers of order one or less. Basically, we observed

only two bulges even at Reynolds numbers of order 100, with

some very rare exceptions. Perhaps the number of bulges on

the ring is related to capillary breakup and is strongly in-

fluenced by the value of interfacial tension. We saw many

bulges when soap was added to water in soybean oil (see the

last entry of Table II). This interpretation is also suggested

by the closely similar instability in miscible liquids,6 in

which case the lack of surface tension promotes the forma-

tion of many more nodules around the ring, the ring breaks

into many drops, and those drops subsequently repeat the

cycle and there is a vortex cascade.5 Surface tension can keep

subsequent drops spherical if they are small enough, and

thus inhibits the cascade.

The effect of the wall on the drop and ring needs further

study. For example, when a ring approaches a wall of the

apparatus, it expands considerably before touching it. Also,

experiments done with a vortex tube of smaller diameter

show that the walls inhibit ring formation. Observations

about the way a ring behaves (in the miscible case) at a

variety of boundaries is reported in Ref. 18.

The dynamics leading to formation of vortex r ings is not

well understood. Data presented in the next section show

that rings form from drops started from rest when the viscos-

ity of the drop is relatively great and the inter-facial forces do

not dominate viscous forces.

A falling drop is relentlessly sheared by the host fluid,

but only small portions of the host fluid come under the

TABLE IV. Systems of injected bubbles.

influence of the falling drop, and these only momentarily. If

we move with the drop, we can think that we have a uniform

flow around the drop as in Fig. 9 (b), and this picture is also

suggestive of why circulations develop in the drop and not in

the host fluid. The flow around the ring would, in the case

where the ring fluid is very viscous, resemble that of Fig.

1 (a), where the flow is analogous to the rigid-body rotation

of Fig. 1 (b) and would then be almost potential flow, with

potential flow at infinity (uniform flow), and the vorticity

localized to the interface region between the fluids.

VII. TWO-FLUID SYSTEMS THAT DO AND DO NOT

FORM VORTEX RINGS

We used formula ( 13) to compute the velocity of a fall-

ing drop [with a = (2.39) 13 ] and rising bubble [with vol-

ume 5 cm3, a = ( 1.19) 13 and evaluated many of the di-

mensionless parameters. The parameters are listed in Tables

II-IV. Parameters that are not set down explicitly in these

tables can be computed readily from the listed values. Table

II tabulates the systems that were observed to form rings.

The other two tables list the two-fluid pairs that were ob-

served not to form rings.

We find that to form a ring, it is necessary that the drop

fluid be much more viscous than the host 3uid. Another

criterion which appears to be necessary is that the effect of

interfacial tension should be smaller than viscous effects,

which may be expressed as

System

100 cS SiVwater

12 500 cS U/water

Soy oil/water

30W motor oil/water

Olive oil/water

S*

u

(dynkm)

f-Q/P

(cmhc)

Rd

M

Jd/&

J/R

24.67

0.96

984

1043

96

0.03

2.88

27.11

0.975

621

5.3

12 188

0.0004

4.8

3.39 0.922

1917

3829

49

0.004

0.196

9.22 b.886

280

16.42

0.914

2119

3257

63

0.012

0.72

576

Phys. Fluids A, Vol. 4, No. 3, March 1992 Baumann eta/.

576


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12/15

FIG. 14. Vortex ring of dyed glycering falling in soybean oil after blowout.

FIG. 13. Crisco rising in a column of water with surfactant (Alconox). The

value of interfacial tension has been reduced from 3.39 dyn/cm to 0.158

dyn/cm by the surfactant. The membrane does not break, despite the low

tension. (a) A torus is formed inside the water bag. (b) The water in the

torus is dragged out in the wake.

J/R


13/15

quantities ( l-7 ml) of 1000 CSsilicone oil in soybean oil, and

allowing them to fall under the influence of gravity. The

results of pure silicone drops, as well as silicone drops con-

taining the surfactant Igepal are included.

1. Introduction

The drops of silicone oil are released at the top of a 4 ft

tall glass tube that contains soybean oil. They fall under

gravity, and observations are made as they traverse the

length of the tube. To control the drop sizes, we use a 1 in.

diameter (60 ml) plastic syringe with the end almost com-

pletely removed. Since small amounts of silicone invariably

stick to the syringe and since the modified syringe volume

was measured to be 0.8 ml less than with the tip intact, we

consistently pull the plunger back 1 ml beyond the desired

drop size. With the plunger set, the syringe is filled with the

silicone oil and the end is quickly placed into the soybean oil

and the plunger is depressed. The syringe is tilted to a sharp

angle and twisted to remove the clinging drop. While this

method does not allow us to ascertain with a great degree of

accuracy the true drop volume, it does provide for drops of

very consistent volume.

(b)b)

FIG. 15. (a) The center-spanning membrane is rupturing while still aiIG. 15. (a) The center-spanning membrane is rupturing while still ai

bottom of this 2 ml drop of pure silicone oil. (b) Here, the membraneottom of this 2 ml drop of pure silicone oil. (b) Here, the membrane

bulged up through the center of the 5 ml, pure silicone drop, forming a 1,ulged up through the center of the 5 ml, pure silicone drop, forming a 1,

dome.ome.


: the

has

arge

I. The drops begins t o indent on top.

II. The indentation deepens. Very oman drops

(less than 1 mU

remain

like this for the entire

length of the tube.

III. The indentation reaches the bottom of the drop.

Drops that are I-2ml will sometimes remain in

thfs mnfi~tion for the length of the tube.

00

N. Membrane rupture occurs on the bottom of the

drop. Those dmps 2-3111 will usually exhibit this

type of rupture.

PIG. 16. Drop evolution for volumes of 2-3 ml.

2. Pure silicone oil

Small drops (roughly those < 2 ml) were generally ob-

served to remain as indented oblate spheres for the entire

length of the tube. When drop sizes were increased to 2 ml,

the indentation in the drop deepened, and for many drops

The drop forms an ablate sphere

An indentation be&s to form on the

top of the ablate sphere, flatt ening it

out, and folring more fluid towards

the outer edge.3 of the drop.

The indent&m becomes deeper. and

still more fluid is forced into the outer

edgsofthedmp

The middle of the drop h now a fairly

thin

membrane,

with a thick annulus

surrounding it.

The membrane is stretched upward, and

becomes very thin. Dmps of 4 ml and

more exhibit Rayleigh-Taylor imtabili ty

at this point and do not proceed to

phase VI.

The membrane breaks, leaving only

the vortex ring. This ccdiguration is

unstable, and the ring will separate

into two lobes shortly after forming.

FIG. 17. Drop evolution for volumes of 3-5 ml.

Baumann eta/.

578


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14/15

(b)

FIG. 18. (a) This 5 ml drop of loo0 cS silicone containing 0.5% Igepal CO-

530 has formed a membrane similar to that in Fig. 15(b). (b) The mem-

brane is rupturing from the left side of the drop to the right. Rupture of

membranes at this stage of development is extremely rare with pure silicone

oil, but occurs frequently when Igepal is added.

(six times out of ten observations) this indentation poked all

the way through until a free ring was generated. These rings

would always rupture their membranes while they were on

the bottom of the drop [see Figs. 15(a) and 161.

Drops of approximately 3 ml also form rings, but their

evolution is slightly different than that of 2 ml drops in that

the spanning membrane does not always break while it is on

the bottom. These drops instead expand horizontally, and

the membrane bulges up through the center of the surround-

ing annulus of fluid [see Figs. 15 (b ) and 171. The degree to

which it bulges upward is strongly dependent on drop size;

the larger the drop, the more extreme the expansion. For

drops with volumes of 3 ml, the membrane would occasion-

ally rupture after this expansion. However, those drops of

pure silicone oil having a volume greater than 3 ml were not

observed to form rings. They underwent the same evolution

as the slightly smaller drops, but exhibited Rayleigh-Taylor

instability before membrane rupture (see Fig. IO). The in-

stability causes the drop to form two lobes, thus pinching off

the center membrane.

TABLE V. Data summary. Quantities listed are pure 1000cSsilicone/1000

cS siIicone with 0.5% Igepal. The interfacial surface tension for the pure

silicone oil in soybean oil is 2.7 dyn/cm. With 0.5% Igepal in the silicone oil,

the interfacial tension is 1 dyn/cm.

Size (ml) # Trials # Ruptures # Pinch-off # Oblates

1.0 5/5 o/o o/o 5/5

2,O

IO/10

6/4 o/o 4/6

2.5 2/4 2/4 o/o o/o

3.0 lO/lO 3/10 7/o o/o

4.0

lO/lO

o/9

IO/l o/o

5.0 2/10 o/7 2/3 o/o

7.0 3/- O/- 3/- O/-

3. Silicone oil with 0.5% lgepal CO-530

The addition of 0.5% Igepal CO-530 to the 1000 CS sili-

cone oil had a rather dramatic effect on the upper bound of

drop size. As mentioned above, large drops (4-7 ml) had the

tendency to become unstable, form a tough membrane and

pinch off before a ring could be formed. The drops which

contained Igepal proceeded in the same fashion except that

the membrane usually broke; sometimes before and some-

times after the onset of instability. It was sometimes ob-

served that a membrane would rupture during pinch off. In

other words, it appeared that the membrane rupture oc-

curred when the total surface area of the membrane was

decreasing. Without Igepal, 4 ml and larger drops would

invariably pinch off without membrane rupture, while with

Igepal, 4 and 5 ml drops would frequently (nine and seven

times out of ten observations, respectively) form free rings

[see Figs. 18(a) and 18(b) 1. Thus large drops with Igepal

experience membrane rupture more frequently than those

without, and so it would appear that the membrane strength

was in some sense decreased with the addition of the Igepal.

The effect of Igepal on smaller drops sizes is not clear.

For 2 ml drops, membranes broke less often with the Igepal

(four out of ten with Igepal, six out of ten without). How-

ever, with 2.5 ml drops membranes broke much more fre-

quently with Igepal (ten out of ten with, three out of ten

without). At 3 ml, no effect was observed when Igepal was

added (membranes broke ten times in ten trials with or with-

out). It appears that with small drops, size is a more critical

factor in determining whether a ring will form than whether

or not the drop contains any Igepal.

The raw data are presented in Table V. Under each col-

umn two numbers are given; the tist is for pure 1000 CS

silicone oil, the second for the same containing 0.5% Igepal.

The column with the heading # Ruptures lists the num-

ber of times the membrane was observed to rupture, #

Pinch-off lists the number of times the instability manifest-

ed itself before the membrane could rupture, and Oblates

lists the number of drops that were observed to remain as

indented oblate spheres for the entire length of the tube.

The number of trials is quite small (ten and less) at each

drop size, and it could certainly be said that more observa-

tions should be made. However, for the most part we found

that given the drop size, we could predict quite accurately

whether a ring would form,

579 Phys. Fluids A, Vol. 4, No. 3, March 1992 Baumann ef d 579


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15/15

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580

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